Conjecture on decreasing missing-sum distributions

Let $R_n”$, $R_n'$ and $R_n$ denote the three random-set models used in the paper, and let $f_{[0,n-1]}^+(R)$ denote the number of missing sums in the interval $[0,n-1]$. For each distribution, let $m^*$ be the value where its maximum is reached.

Missing-sum monotonicity conjecture. For $m\geq m^*$, the following assertions hold:

1. $P\left[f_{[0,n-1]}^+(R_n”)=m\right]$ is decreasing.
2. $P\left[f_{[0,n-1]}^+(R_n')=m\right]$ is decreasing except for $n=4,5,6$.
3. $P\left[f_{[0,n-1]}^+(R_n)=m\right]$ is decreasing except for $n=1$.

These assertions formalize the observation that the distributions for the number of missing sums may be decreasing in the range after their maximum. The source presents this as a conjecture based on the preceding theorem and discussion, with no proof or resolution given.